1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

276 8. AF Embeddability

Proof. Let I c B be an inclusion such that B is AF and let {en} c I be an

approximate unit of I. Let J c B be the hereditary subalgebra generated

by I (i.e., the norm closure of the algebras enBen)· Evidently {en} is also an

approximate unit for J so we only need to recall that hereditary subalgebras

of AF algebras are again AF.

This fact is standard fare so we only sketch the proof. The hardest part

is to show that if J c B is hereditary, then J has an approximate unit of

projections. (The informed reader may recall that this fact even holds when

B is only assumed to have real rank zero.) Once this is established, the

proof is routine. Indeed, for any finite set ~ c J and r:; > 0 we can choose

a projection p E J such that llpx - xii < r:; for all x E ~. Then we can

find a finite-dimensional subalgebra C c B which almost contains {p} U ~.

Perturbation theory yields a projection q E C which is close to p and this

implies the existence of a unitary u E B which is close to the identity and

such that uqu* = p. One then checks that u(qCq)u* is a finite-dimensional

subalgebra of J which almost contains~. See [64] for the details. D

We now prove some results on the AF embeddability of extensions.

Lemma 8.4.5. Let 0 ---+ I ---+ E ---+ B ---+ 0 be exact where I <1 E is essential

and I is AF embeddable. Then there exists a commutative diagram

0-----+ I -----+ E -----+ B-----+ 0

l l l idB

0-----+ J -----+ F -----+ B-----+ 0

where J is AF and essential in F, the bottom row is exact, the vertical

arrows on the left and in the middle are injective, and the vertical arrow on

the right is the identity.

Proof. Apply the previous lemma and we have E c M(I) c M(J). Since

M(I) n J = I, it follows that En J = I and so we define F = E + J to

complete the proof. D

Here is (a special case of) the result of Kasparov that we will need (cf.

[97]).

Theorem 8.4.6. Let I C IIB(H) be a nondegenerate representation of a
separable nuclear C -algebra, B be separable and CJ: B ---+ M(JK(JC) @I) C
IIB(JC@H) be any -homomorphism. Ifw: B---+ IIB(JC)@Cl c M(JK(JC)@I) c
IIB(JC@ 7-l) is any faithful representation such that w(B) n JK(JC) = 0, then
there exists a unitary operator U : (JC @ 7-l) EB (JC @ 7-l) ---+ JC @ 7-{ such that

w(b) - U(w(b) EB CJ(b))U* E JK(JC) @I,